He was born in Cremona in Lombardy, then a part of the Austrian Empire, and now part of Italy. Beltrami first began studying mathematics at University of Pavia in 1853, but was forced to discontinue his studies in 1856 because of financial hardship. He was appointed to the University of Bologna as a professor in 1862, the year he published his first paper. Beltrami later taught at universities in Pisa, Rome and Pavia. He died in Rome in 1899.
In 1868, (in Essay on an interpretation of non-Euclidean geometry) Beltrami gave the first model of hyperbolic geometry. In Beltrami's model, lines of hyperbolic geometry are represented by geodesics on the pseudosphere. Thus, Beltrami attempted to prove that Euclid's parallel postulate could not be derived from the other axioms of Euclidean geometry; this proof fails however since the pseudosphere is only a small portion of the hyperbolic plane.
In the same year, however, Beltrami went much farther and gave a correct proof of the equiconsistency of hyperbolic and Euclidean geometry, by defining what are now known as the Klein model, the Poincaré disk model, and the Poincaré half-plane model, in his paper Teoria fondamentale degli spazii di curvatura costante. For the half-plane model, Beltrami cited a note by Liouville in a book by Monge on differential geometry.